Mathematics - 2017-08-22 (page 1 of 4)

Daminark

00:02

"Hello sign errors my old friend"

Lucas Henrique

00:20

hey

Daminark

Hey @LucasHenrique, it's been a while

Lucas Henrique

Hello @Daminark!

I've been a bit busy because of my exams

How are you doing?

Daminark

I'm doing alright, thanks! Right now I'm doing summer camp

ATaco

00:38

I'm working on a MathQuill Latex Editor for ChatJax

Akiva Weinberger

03:54

Guys

Is a male mattress called a matter

Daminark

Absolutely

ALannister

04:19

Whoa, looks like I've had this window open all this time and I've been missing all the fun.

Leaky Nun

ughhh how do I prove that if y' = y + y^3 is zero anywhere then it is zero everywhere?

ATaco

I've got my MathQuill editor working.

Leaky Nun

nice @ATaco

I know that I can use definite integration for this case, but what if the bernoulli differential equation is more complicated...

ATaco

04:39

Oh, I just noticed it's you Leaky. :P Didn't recognize you with an avatar.

Secret

05:24

[Random to be expanded] Changing lanes and galliean relativity

Jon

05:35

That is allowed

Nevermind

Taking linear algebra II this semester, no calculator, and notes allowed on exam? Your thoughts?

Perturbative

Hey everyone!

Jon

Good morning friend

Secret

06:29

[Random]

When whole commutative diagrams are functors

en.m.wikipedia.org/wiki/Diagram_(category_theory)

ln x + ln y = ln (x+y) => xy = x+y => xy - x - y = 0 => x(y-1) - y = 0 => x(y-1) = y => x = y/(y-1)

Kirill

07:40

@Mr.Xcoder yes, you have two steps: 1) start, and 2)step. Start with $n=0$ or $n=1$ - depending on what you call natural numbers. If the theorem is true for $n=1$ or $n=0$ - good, you've had a perfect start. Now you say that the theorem is true for some $n$ - not for all, just for some. Prove that the theorem is true for $n+1$. Use what you have proven for $n=0$ or $n=1$. Look at some exercises showing you how do to that correctly and try to prove some things yourself.

Daminark

Hey @AlessandroCodenotti and @SteamyRoot!

Tobias Kildetoft

@Daminark Hi

Daminark

Oh hey @TobiasKildetoft! How's it going?

Tobias Kildetoft

good

Daminark

Oh btw, when do you have to submit your grant application?

Tobias Kildetoft

07:46

September 15th

Daminark

Ah, hopefully that'll be enough time

Also hey @WillHunting!

Tobias Kildetoft

It should be. The main part of the project description is done now, I just need to tweak stuff

Alessandro Codenotti

Hi @Dami

Will Hunting

@Daminark Hello!

@Semiclassical Thanks, I think the lyrics are fitting in some sense.

Daminark

Sick

How's it going guys?

Will Hunting

07:51

@Jon It's normal to have no calculator and no notes. In fact, that should be the way an exam goes. All those exams that require calculators and notes are stupid.

Mr. Xcoder

@Kirill already did that. Thanks for your suggestion!

Mr. Xcoder

08:49

Hi @LeakyNun

Leaky Nun

Is $S_6 \oplus S_7$ isomorphic to $S_{10}$?

Guten Tag @Mr.Xcoder

Tobias Kildetoft

@LeakyNun You mean those to be the symmetric groups?

Leaky Nun

Bayern? @Mr.Xcoder

@TobiasKildetoft yes

Tobias Kildetoft

And the $\oplus$ should be a $\times$ then?

The first is not correct to use for non-abelian groups

Leaky Nun

@TobiasKildetoft yes

Tobias Kildetoft

08:51

Then no, since the direct product will have more normal subgroups than $S_{10}$.

Mr. Xcoder

@LeakyNun Umm. What?

Leaky Nun

@TobiasKildetoft can we analyze the order of elements instead?

Tobias Kildetoft

@LeakyNun Sure, that will work too

Leaky Nun

@Mr.Xcoder "i mag ihre Deutsch" looks Bavarian to me

@TobiasKildetoft how?

Mr. Xcoder

@LeakyNun Ah, yeah, that's it

Tobias Kildetoft

08:53

@LeakyNun The direct product will have elements of larger orders than $S_{10}$

Leaky Nun

so you speak Bavarian German?

@TobiasKildetoft like 5 times 7 = 35?

Mr. Xcoder

@LeakyNun No, that is the only phrase I know in Bavarian German

Tobias Kildetoft

@LeakyNun Yeah

Leaky Nun

@Mr.Xcoder oh, scheen

@TobiasKildetoft isn't it amazing that the orders of the two groups are the same

Mr. Xcoder

@LeakyNun What does "scheen" mean?

Leaky Nun

08:55

@Mr.Xcoder schön

Mr. Xcoder

Oh

Tobias Kildetoft

@LeakyNun Not particularly

Leaky Nun

@TobiasKildetoft heh

Mr. Xcoder

BTW I learned what "Induction" is

Leaky Nun

@Mr.Xcoder good for you

Mr. Xcoder

09:00

Is $0$ considered a natural number?

Tobias Kildetoft

@Mr.Xcoder Depends on the context

Mr. Xcoder

Oh, there is no consensus on that?

Gotta go now. I saw an answer on M.SE on that. Bye o/

Tobias Kildetoft

none at all

Will Hunting

10:37

@TedShifrin Yes, there is short term evolution. When you fart, you evolve gas for a short time.

user302782

Hi

user302782

I do not know why I got two down votes for my answer in this question: math.stackexchange.com/questions/2401960

user302782

Is there a way that I acn find the reason for the down votes?

Will Hunting

@VafaKhalighi There are many crazy people on this site who downvote for no reason.

@VafaKhalighi No, there is no way. I have learnt that votes on this site mean nothing. You can ask about your answer, which I haven't read, but don't try to make sense of votes.

user84215

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Will Hunting

10:46

Just like there is a way to ignore people in chat, there should be a way to ignore people on the main site too, with a simple mouse click!

Leaky Nun

@VafaKhalighi there is only one

user84215

This is one of my opinions in my deleted posts on the Meta MSE.

user302782

@WillHunting Thanks for your answer. When I got the down votes, I thought that there might have been problems with my answer and I was only curious to see what was the math problem in my answer. I think if the votes become constructive, it helps a lot to improve. Also if one down votes an answer, it would not harm to write a reason for voting down otherwise how does one improve?

user302782

@LeakyNun The actual question page shows 1 down vote but my own profile shows -2

Will Hunting

@VafaKhalighi Exactly. I won't read your answer now, but there are plenty of right answers that get downvotes and wrong answers that get upvotes on SE in general, because almost anyone can cast a vote.

user84215

10:52

This fact is also true in the Meta MSE.

user84215

Unfortunately

Will Hunting

@VafaKhalighi A downvote costs you 2 points. That answers the question.

Leaky Nun

@VafaKhalighi your answer can be improved by actually solving for $g^{(n)}(x)$.

user302782

@WillHunting Thanks for the explanation.

user302782

@LeakyNun Thanks for the feedback. I try to edit my answer with improvements when I get some spare time.

user84215

10:56

Please do not pursue these things. Your account may be suspended.

user302782

@MathematicsAminPhysics Why does my account get suspended by asking what I did wrong?

Will Hunting

@VafaKhalighi It doesn't get suspended. He's talking nonsense.

user84215

As they have said, some users downvote and upvote for no reason. Suspending your account may be happened for no reason.

Martin Sleziak

@MathematicsAminPhysics Now do you mean post ban (question ban or answer ban) or do you mean suspension?

user84215

both of them

Martin Sleziak

11:02

Because downvotes are related to question ban and answer ban, but they are not related to account suspensions.

user84215

How about pursuing unfair downvoting?

Martin Sleziak

Many users asked about down-votes on their post on meta and it did not result in suspension.

There is even a separate chatroom which was created for this purpose (among other things) - the rom description says: "You can get feedback for questions, answers and comments."

Will Hunting

@MartinSleziak His account was suspended for a week for something we don't know. I have decided not to talk to him anymore, because everything he says seems quite trollish and cranky.

user84215

You do the best.

user84215

Please ignore me.

user302782

11:06

@MathematicsAminPhysics I do not get your logic for suspending account. On the other hand, I have had no complaints about the down vote; as you can see, I was only curious to find out what I did wrong so I could improve in my future answers. I can not think why would that make my account being suspended?

Will Hunting

@VafaKhalighi Like I said, your account won't be suspended for that. He's talking absolute nonsense.

user84215

@VafaKhalighi Ok. If you have no complaint, your account will not be suspended.

Mr. Xcoder

Mods won't suspend any account without a very good reason. Asking why a downvote occurred is a very good question. In fact, anyone that downvotes should comment with their reasoning.

2

Will Hunting

@Mr.Xcoder In fact, usually, even before a suspension occurs, there will be dialogues with the user.

user84215

@Mr.Xcoder ^^^^ To your second sentence.

Mr. Xcoder

11:10

@WillHunting I know the process.

user84215

For me there was no dialogue.

Mr. Xcoder

They ~~usually~~ always send at least one moderator message before suspending an account, that explains their reasons..

2

user84215

No.

Akiva Weinberger

"Yields falsehood when preceded by its quotation" yields falsehood when preceded by its quotation.

Leaky Nun

@AkivaWeinberger mind=blown

"Yields falsehood when preceded by its quotation" yields falsehood when preceded by its quotation.
> ["Yields falsehood when preceded by its quotation" yields falsehood when preceded by its quotation] is false.
> "Yields falsehood when preceded by its quotation" yields truth when preceded by its quotation.
> ["Yields falsehood when preceded by its quotation" yields falsehood when preceded by its quotation] is true.
> "Yields falsehood when preceded by its quotation" yields falsehood when preceded by its quotation.

Alessandro Codenotti

11:21

@LeakyNun Hofstadter?

Leaky Nun

@AlessandroCodenotti ?

Alessandro Codenotti

I was guessing the author of that quote

I've seen it before somewhere but I don't remember where

Leaky Nun

how would i know

Alessandro Codenotti

Oh, I wanted to answer to Akiva's message

I hate using this chat from my phone

Tim The Enchanter

Just a quick conjecture, is any limit point of the set $S = \left\{ \frac{1}{n} + \frac{1}{m} : m,n \in \mathbb N \right\}$ of the form $\frac{1}{k}$ for some $k \in \mathbb N$?

Leaky Nun

11:30

I second your conjecture

Akiva Weinberger

11:41

https://en.wikipedia.org/wiki/Quine's_paradox

@AlessandroCodenotti @LeakyNun

Incidentally, a "quine" is a computer program that outputs its own code.

Tim The Enchanter

Nvm I just found it on M.S.E math.stackexchange.com/questions/930646/….

Martin Sleziak

@TimTheEnchanter [Approach0] says yes. Find the derived set of $\{\frac{1}{n} + \frac{1}{m}: m,n \in \mathbb{N}\}$ and prove it is such. and find the number of limit points of the set{$ \frac{1}{m} +\frac{1}{n}:m,n \in \Bbb N$}

Sorry once again with links: Find the limit points of the set $\{ \frac{1}{n} +\frac{1}{m} \mid n , m = 1,2,3,\dots \}$ and find the number of limit points of the set{$ \frac{1}{m} +\frac{1}{n}:m,n \in \Bbb N$}.

Found using this Approach0 search.

But I do not doubt that there are many other posts about this on the main.

Mr. Xcoder

I have a problem and I have absolutely no idea how to go about it: Let $A = \{\sqrt{3}-\sqrt{2}\}+\{\sqrt{4}-\sqrt{3}\}+\dotsb+\{\sqrt{200}-\sqrt{199}\}$, where $\{x\}\text{ is the fractional part of }x$. What is $\lfloor A\rfloor$? We also have choices: a) 50, b) 12, c) 13, d) 49

@LeakyNun Can you help?

Akiva Weinberger

"is a sentence fragment" is a sentence fragment

- Wikipedia

SteamyRoot

RIP me

Leaky Nun

11:45

@Mr.Xcoder inconsistent sign

Mr. Xcoder

Oh sorry, correcting

@LeakyNun Fixed

Absolutely no clue

Leaky Nun

@Mr.Xcoder prove that each term is less than 1

Mr. Xcoder

@LeakyNun Well, of course each term is smaller than one since it's the fractional part

SteamyRoot

(non-maths) student solves the differential equation $\frac{1}{\sin(2t)} x'(t) + x(t) = \frac{1}{2}x^2(t)$ by substituting $y = x^{-1}$, and finds (correctly) that $y(t) = \frac{1}{2}\left(1 + c\exp\left( -\frac{1}{2} \cos(2t)\right)\right)$

Leaky Nun

@Mr.Xcoder I mean, inside the fractional part

Akiva Weinberger

11:48

@Mr.Xcoder No, the stuff inside it

Snaped

Mr. Xcoder

OOhhh.... Let me think

SteamyRoot

"Therefore $x(t) = \frac{1}{y(t)} = 2 + \frac{2}{1+c\exp(...)}$"

WHYYYYYYYYY

Mr. Xcoder

@LeakyNun How?

Leaky Nun

@Mr.Xcoder prove that $\sqrt3-\sqrt2<1$

@SteamyRoot I call that "not even wrong"

to be fair, you did say that said student is non-maths

SteamyRoot

Wait, I typoed it again. It should've been 1/2 + 1/c exp(...)

whatever

Well, yeah, but the idea is like... You solve a pretty damn difficult differential equation...

Akiva Weinberger

11:51

Oh I see a quite simple way to do that (re prove it's less than 1)

Leaky Nun

@AkivaWeinberger of course it's simple

Akiva Weinberger

Without resorting to concavity or anything

which was equivalent to my first idea

SteamyRoot

And then you go all 1/(a+b) = 1/a + 1/b

Leaky Nun

@AkivaWeinberger in that case it would be too complicated

Mr. Xcoder

$\sqrt{3}-\sqrt{2}<1 | \text{^2}$, $(\sqrt{3}-\sqrt{2})^2<1, \implies 3-2\sqrt{6}+2<1$

Akiva Weinberger

11:53

Squaring it was an idea too but I couldn't finish the thought in my head so I abandoned it

(I, uh, don't have pencil and paper handy)

Leaky Nun

@Mr.Xcoder begging the question fallacy

@AkivaWeinberger you just have to square it the right way

Akiva Weinberger

@LeakyNun Yeah, but it's positive, so that's really an $\Leftrightarrow$

so it's technically OK

Mr. Xcoder

Hint to get on the right track?

Akiva Weinberger

@LeakyNun (Do we actually have the same idea?)

Leaky Nun

@AkivaWeinberger he explicitly wrote $\implies$

Akiva Weinberger

11:54

@LeakyNun Yeah but still

Whatever

Leaky Nun

@AkivaWeinberger no, I don't need to square it to prove it

Akiva Weinberger

OK?

SteamyRoot

$\sqrt{4} > \sqrt{3} > \sqrt{2} > \sqrt{1}$ and $\sqrt{4} - \sqrt{1} = 2- 1 = 1$

Mr. Xcoder

Well, $|\sqrt{3}-\sqrt{2}|=\sqrt{3}-\sqrt{2}$

Akiva Weinberger

I mean I'll say the thing if you're OK with that @LeakyNun

It's just one word really

Leaky Nun

11:55

it's one word

but don't say it

Akiva Weinberger

OK sure

Mr. Xcoder

Godammit, I have to go. Will be back shortly

Akiva Weinberger

How was the eclipse?

Actually I don't remember where you are

Leaky Nun

@AkivaWeinberger hong kong

Akiva Weinberger

If you're in Hong Kong you didn't see an eclipse

SteamyRoot

11:56

2

Leaky Nun

@AkivaWeinberger right

Akiva Weinberger

(Sniped)

Leaky Nun

@SteamyRoot hairgenesis?

$\begin{array}{rcl}
\dfrac1{\sin(2t)} \dfrac{\mathrm dx}{\mathrm dt} + x &=& \dfrac12x^2 \\
\dfrac1{\sin(2t)} \dfrac{\mathrm dx}{\mathrm dt} &=& \dfrac12x^2 - x\\
\dfrac2{x^2-2x} \dfrac{\mathrm dx}{\mathrm dt} &=& \sin(2t)\\
\displaystyle \int \dfrac2{x^2-2x} \dfrac{\mathrm dx}{\mathrm dt} \ \mathrm dt &=& \displaystyle \int \sin(2t) \ \mathrm dt\\
\displaystyle \int \dfrac2{x^2-2x} \ \mathrm dx &=& \displaystyle \int \sin(2t) \ \mathrm dt\\
\displaystyle \int \left(\frac1x-\frac1{x+2}\right) \ \mathrm dx &=& \displaystyle \int \sin(2t) \ \mathrm dt\\

@SteamyRoot amirite

SteamyRoot

Looks like it definitely could be

Alex K Chen

How do I create olympiad combinatorics simulator in python ? e.g To solve IMO 2010 P5 (See here), you create a python applet like this: where you can push buttons and gain intuition ?

Leaky Nun

12:01

inb4 definite integrals

SteamyRoot

The absolute values seem like they might be troublesome, though. I use a different approach

Alex K Chen

artofproblemsolving.com/community/c6h356196p1936917

Leaky Nun

@AlexKChen just use console :P

Alex K Chen

What's console ?

Leaky Nun

@AlexKChen cmd

Alex K Chen

12:03

Graphical handing are bit better.

Leaky Nun

use tkinter for graphical

for a demonstration of console interaction:

while 1:n=int(input("Enter a number: "));print(n**2)

if you're mostly doing that for yourself, then a console-interaction would be easier to program

Alex K Chen

Oh, so learning tkinter is sufficient for doing the grpahics stuff ?

Leaky Nun

@AlexKChen yes

wiki.python.org/moin/TkInter

Alex K Chen

Also, can I do most Simon Tatham's game on tkinter ?

Leaky Nun

@AlexKChen I think so

I've never used TkInter, as a disclaimer

Alex K Chen

12:06

Oh OK.

Mr. Xcoder

@LeakyNun I'm back

Can you give me a hint?

Leaky Nun

@Mr.Xcoder try more first

Mr. Xcoder

@LeakyNun Ok. So I should abandon the "square approach"?

Or square it another way?

Leaky Nun

@Mr.Xcoder all roads lead to rome

@AlexKChen here's a console-interaction program I wrote:

Mr. Xcoder

Fancy poetry

Leaky Nun

12:13

coins = [1]*6
while True:
	print("coins: %s" % coins)
	allowed = [(1,i) for i in range(1,6) if coins[i-1] != 0] + [(2,i) for i in range(1,5) if coins[i-1] != 0]
	print("allowed operations: %s" % allowed)
	op = 0
	while op not in allowed:
		try: op = eval(input("input operation: "))
		except: pass
	if op[0] == 1:
		coins[op[1]-1] -= 1
		coins[op[1]] += 2
	else:
		coins[op[1]-1] -= 1
		coins[op[1]], coins[op[1]+1] = coins[op[1]+1], coins[op[1]]

as you see, it can be done in very few lines of code

Mr. Xcoder

Let me do it by hand, thinking in MathJax is hard

Alex K Chen

@LeakyNun Why it's showing "No module named tkinter ?"

Leaky Nun

@Mr.Xcoder you may not be able to find the answer. that isn't the point. at least you have explored in the vast world of mathematics, instead of relying on others to give you the answer

Mr. Xcoder

@AlexKChen Because you must install it first

Leaky Nun

@AlexKChen what did you do?

@AlexKChen you need to do python -m pip install tkinter

Alex K Chen

12:20

I wrote import tkinter as tk in IDLE

@LeakyNun Like what I did the other day ?

Leaky Nun

@AlexKChen yes

Mr. Xcoder

@LeakyNun What do you mean by at least you have explored in the vast world of mathematics, instead of relying on others to give you the answer?

Leaky Nun

@Mr.Xcoder exploration...

trying out different methods

failing

as they say in Chinese

Mr. Xcoder

I have no other methods in mind, but I'll seek one

Leaky Nun

failure is the mother of success

mathematics is not about the answer

not about finding the correct answer

maths isn't about finding the correct answer, it's about finding the correct answer

Mr. Xcoder

12:24

unlike physics

Alex K Chen

Mr.Xcoder is surly a troll.

Mr. Xcoder

@AlexKChen i don't think so

Alex K Chen

@Mr.Xcoder I don't think that, too.

@LeakyNun It's stuck at "downloading/unpacking tkinter"

Now showing:

No distribution at all found for tkinter

In red ink.

Leaky Nun

@AlexKChen no, he isn't

Alex K Chen

OK I was joking :P

Why the error ^^

Mr. Xcoder

12:27

@LeakyNun I have proven that $\sqrt{x+1}-\sqrt{x}<1, \forall x \ge 2$.

I think it's ok

Leaky Nun

@AlexKChen wait a minute, I'm figuring it out

@Mr.Xcoder how did you prove it?

Mr. Xcoder

by squaring

wait a bit

Leaky Nun

@AlexKChen you don't need to install it; it's already inside python

what is your python version?

Alex K Chen

2.7.9

Leaky Nun

@AlexKChen OS?

Mr. Xcoder

12:32

@LeakyNun I found a mistake in my proof... Rewriting

Alex K Chen

Windows 8.1

Leaky Nun

for python 2.7.9 it is Tkinter not tkinter apparently

Alex K Chen

Damn $h!t

Yeah works now.

Mr. Xcoder

@LeakyNun I must specify from the beginning that $x\ge 1$. $\sqrt{x+1}-\sqrt{x}<1\leftrightarrow (x+1)-2\sqrt{x^2+x}+x<1\leftrightarrow \\ 2x+1-2\sqrt{x^2+x}<1\leftrightarrow 2x-2\sqrt{x^2+x}<0\leftrightarrow 2x<2\sqrt{x^2+x}\implies x<\sqrt{x^2+x}$. $\sqrt{x^2+x} > x, (\forall x\ge 1)\implies x<\sqrt{x^2+x} \text{ is true.}$ That implies our initial relation, $\sqrt{x+1}-\sqrt{x}<1$ holds as well $\forall x\ge 1$. Correct? I think I have been struggling a bit, not 100% sure it's correct.

Leaky Nun

@Mr.Xcoder do you want to know two more methods to prove it?

Mr. Xcoder

12:43

@LeakyNun Yes, I'd be interested in a shorter and maybe more complete approach. I'd also like to know if mine is valid.

Leaky Nun

yes, your approach is valid

Mr. Xcoder

Thanks. I'll let you write / hint to the other 2.

Leaky Nun

first method: move $\sqrt2$ to the right hand side before squaring both sides
second method: conjugate

Mr. Xcoder

I don't know what conjugates are though

Akiva Weinberger

@Mr.Xcoder Take $\sqrt3-\sqrt2<1$ (what you're trying to prove)

Multiply both sides by $\sqrt3+\sqrt2$

What do you get

Mr. Xcoder

12:47

@LeakyNun $\sqrt{3}<\sqrt{2}+1\leftrightarrow 3<2+2\sqrt{2}+1\leftrightarrow 0<2\sqrt{2}\leftrightarrow 0<\sqrt{2}$. Correct?

Akiva Weinberger

@LeakyNun (Re: the first one) Oh nice

Mr. Xcoder

@AkivaWeinberger Hm, I proved that in 2 different ways with help from Leaky. Can we get back to the question?

Akiva Weinberger

Yeah OK so they're all less than 1 so

Leaky Nun

@Mr.Xcoder the conjugate of a-b is a+b

Mr. Xcoder

@LeakyNun Now I know all are smaller than $1$. No idea what to do next

Leaky Nun

12:49

@Mr.Xcoder first part: prove they are all less than 1 (done)

Akiva Weinberger

$\rm\{thing\}=thing$

Leaky Nun

second part: prove they are all greater than or equal to 0

@AkivaWeinberger wait, not so fast

Akiva Weinberger

@LeakyNun Well that's obvious

Can we pull up the question again though for a second

Leaky Nun

1 hour ago, by Mr. Xcoder

I have a problem and I have absolutely no idea how to go about it: Let $A = \{\sqrt{3}-\sqrt{2}\}+\{\sqrt{4}-\sqrt{3}\}+\dotsb+\{\sqrt{200}-\sqrt{199}\}$, where $\{x\}\text{ is the fractional part of }x$. What is $\lfloor A\rfloor$? We also have choices: a) 50, b) 12, c) 13, d) 49

Akiva Weinberger

Thanks

Yeah OK so they're all between 0 and 1, so the floor of the thingies are the thingies

Mr. Xcoder

12:50

@LeakyNun Well, $(\forall x\ge 1): \sqrt{x+1}>\sqrt{x}\implies \sqrt{x+1}-\sqrt{x}>0$

Akiva Weinberger

Ya

Mr. Xcoder

@LeakyNun Why greater than or equal to 0 and not greater than 0?

Leaky Nun

@Mr.Xcoder because of what we are going to do next

third part: prove that each term is equal to what it inside its bracket

Mr. Xcoder

And now, obviously, since ($\sqrt{x+1}-\sqrt{x})\in(0;1)\implies \{\sqrt{x+1}-\sqrt{x}\}=\sqrt{x+1}-\sqrt{x}$

@LeakyNun Proven ^?

Leaky Nun

@Mr.Xcoder yes

now continue on your question

Mr. Xcoder

12:54

Mhmmm

@LeakyNun Thanks for all your help!

Leaky Nun

@Mr.Xcoder no problem. thanks to @AkivaWeinberger also

Mr. Xcoder

oH yes

@AkivaWeinberger Thanks for all your help!

Akiva Weinberger

You're welcome

(I don't know why you wanted to drag out that last step but OK)

Secret

$\require{mhchem}$

$\ce{H_3O^+}$

It works

Mr. Xcoder

@Secret Hydronium Ion?

Secret

12:59

yup

in The h Bar, 4 hours ago, by Slereah

Oh can we import libraries now?

Time to begin the chemistry invasion lolololololol

Transcript for

$Mathematics$ Mathematics